Two conjectures on the arithmetic in R and C

@article{Tyszka2010TwoCO,
  title={Two conjectures on the arithmetic in R and C},
  author={Apoloniusz Tyszka},
  journal={Math. Log. Q.},
  year={2010},
  volume={56},
  pages={175-184}
}
Let G be an additive subgroup of C, let Wn = {xi = 1, xi + xj = xk : i, j, k ∈ {1, . . . , n}}, and define En = {xi = 1, xi + xj = xk, xi · xj = xk : i, j, k ∈ {1, . . . , n}}. We discuss two conjectures. (1) If a system S ⊆ En is consistent over R (C), then S has a real (complex) solution which consists of numbers whose absolute values belong to [0, 22 n−2 ]. (2) If a system S ⊆ Wn is consistent over G, then S has a solution (x1, . . . , xn) ∈ (G ∩Q) in which |xj | ≤ 2n−1 for each j.